There is a straightforward technique for writing unambiguous recursive descent parsers. The high level algorithm is this: parsing always consumes one character, never backtracks and is locally unambiguous.

You then construct the parser by combining unambiguous parsers from the bottom up. The result ends up unambiguous by construction.

This high level algorithm is much easier to implement without a global lexer. Global lexing can be a source of inadvertent ambiguity. Strings make this obvious. If instead, you lex in a context specific way, it is usually easy to efficiently eliminate ambiguities.